Motion Control Techniques for Synchronous Motions of Translational and Rotary Axes

Available online at www.sciencedirect.com

Procedia CIRP 1 (2012) 265 – 270

5th CIRP Conference on High Performance Cutting 2012

Motion Control Techniques for Synchronous Motions of Translational and Rotary Axes Ryuta Satoa,*, Masaomi Tsutsumib a

Kobe University, 1-1, Rokko-dai, Nada, Kobe, 657-8501, Japan Tokyo University of Agriculture and Technology, 2-24-16, Nakamachi, Koganei, Tokyo, 657-8501, Japan * Corresponding author. Tel.: +81-78-803-6481; fax: +81-78-803-6155.E-mail address: [email protected]. b

Abstract This study proposes motion control techniques which can improve dynamic synchronous accuracy between translational and rotary axes in five-axis machining centres. The proposed techniques consist of servo gain tuning, feed forward controller design, signal delay and backlash compensator, rotational fluctuation compensator and jerk limited acceleration process design. In order to evaluate the effect of proposed techniques, experimental tests are carried out. An experimental apparatus consists of X, Y, and C axes is provided for the experiments. Any control algorithm can be implemented into the experimental apparatus because the system is controlled by a personal computer with a DSP board. Translational X and Y axes are powered by AC servo motors and ball screws. A rotary C axis is powered by an AC servo motor, spur and worm gears. Three kinds of synchronous motions are applied to evaluate the dynamic synchronous accuracies. They are; non-uniformed simultaneous 3-axis motion, simultaneous 2-axis motion of X and C axes, and ellipse shape machining motion using simultaneous 3-axis motion. It is clarified from the experiments various factors including signal delay of rotary encoders influence the dynamic synchronous accuracy between translational and rotary axes, and the proposed motion control techniques can significantly improve the dynamic synchronous accuracy without velocity limitations. 2012 The Published byPublished Elsevier BV. Selection and/or peer-review under responsibility of Prof. Konrad Wegener © 2012 Authors. by Elsevier B.V. Selection and/or peer-review under responsibility of Professor Konrad Wegener Keywords: Machine, Dynamic, Accuracy, Control, Compensation ;

1. Introduction Five-axis centers make a key role in industrial fields to manufacture complex parts, such as impellers. In the machining of complex parts, synchronized multi-axis motions are required. Hence the synchronous accuracy of translational and rotary axes is one of the important factors in the machines. Synchronous accuracy of rotary and translational axes has been investigated up to now. Kakino et al. tried to evaluate the motion accuracy of five-axis machining centers using the ball-bar about 20 years ago [1]. Tsutsumi and Saito measured the synchronous accuracies of translational and rotary axes by the ball-bar in simultaneous 3 of 4 axes motions, and proposed identification methods for geometrical deviations based on the measured results [2][3]. Weikert [4] also measured the synchronous accuracy of translational and

rotary axes, by using the R-test [5]. It is also proposed that the identification method for geometrical deviations based on the measure results of the R-test by Bringmann and Knapp [6]. On the other hand, since the velocity of the axes are rapidly changed in the complex shape machining using simultaneous multi-axis motions, dynamic synchronous accuracy is strongly required. From this point of view, an evaluation method for the dynamic synchronous accuracy between a translational axis and a rotary axis is proposed by one of the authors [7]. The authors also investigated the dynamic synchronous accuracy of translational and rotary axes under non-uniform velocity motions [8], and it is clarified that the synchronous accuracy is deteriorated due to miss match of the dynamic response and servo delay of the axes. This study proposes motion control techniques that can achieve higher dynamic synchronous accuracies.

2212-8271 © 2012 The Authors. Published by Elsevier B.V. Selection and/or peer-review under responsibility of Professor Konrad Wegener http://dx.doi.org/10.1016/j.procir.2012.04.048

Ryuta Sato and Masaomi Tsutsumi / Procedia CIRP 1 (2012) 265 – 270

266

They are; servo gain tuning method, feed forward controller, signal delay and backlash compensator, rotational fluctuation compensator, and jerk limited acceleration process design. Effectiveness of the proposed techniques will be confirmed through the experiments. 2. Experimental Apparatus Figure 1 illustrates the experimental apparatus used in this study. The apparatus consists of an X-Y table (Xaxis and Y-axis), a rotary table (C-axis), a personal computer with a DSP board, and servo amplifiers. The rotary table is attached onto the X-Y table. The whole system is controlled by the personal computer with a DSP board, and it is possible to implement any control algorithm and compensators. Each translational axis is powered by an AC servo motor and a ball-screw, and linear ball guides are used. Lead of the ball-screw is 10 mm, and the stroke of the each axis is 400 mm. Mass of X-axis is 240 kg and 520 kg for Y-axis. The rotary axis is powered by an AC servomotor, spur gears, and a worm gear. Reduction ratio of the spur gear is 4/5, and reduction ratio of the worm gear is 1/72. Hence the total reduction ratio between motor axis and table becomes 1/90. Positional displacements along X and Y axes can be detected by linear encoders with the resolution of 0.1 Pm m. Angular displacement of the rotary table can also be detected by a rotary encoder with the resolution of 0.0001º. Typically, ball-bar or R-test systems are applied to evaluate the synchronous accuracies. Those measurement methods can evaluate both of influences of the static geometrical deviations and dynamic synchronous accuracies. In this study, however, all experimental results in are measured by the linear and rotary encoders, in order to discuss only dynamic synchronous accuracies.

Typical servo system in machine tools has cascade type feedback loop, velocity and position loops. In this study, both of velocity and positional controllers are implemented in the personal computer. Positional controller is proportional (P) controller, and velocity controller is proportional-integral (PI) controller. Torque command signals for each axis are output as voltage signals from DA converters of the DSP board, and the signals are input to the servo amplifiers. Full closed loop position control is executed for the translational axes based on the detected positional displacement by the linear encoders. For the rotary axis, in order to investigate the influences of signal delay and rotational fluctuation, semi closed loop angle control is executed based on the rotational angle of the motor. The rotary encoder attached to the rotary table is used only for measurement of the rotational displacement of the table. 3. Motion Control Techniques 3.1. Servo gain tuning Figure 2 shows the block diagram of proposed controller implemented into the DSP board. Where, TrefX, TrefY, and TrefC are the torque command for the amplifiers. Servo gains; positional loop proportional gain Kpp, velocity loop proportional gain Kvp, and velocity loop integral gain Kvi strongly influence the dynamic responses of the axes, and it is clarified that the synchronous accuracy is deteriorated due to miss match of the dynamic response and servo delay of the axes. Hence the servo gains have to be tuned carefully and adequately to achieve higher synchronous accuracies. From this point of view, in order to avoid the miss match of dynamic response, servo tuning method based X-axis controller

K ff (s )



Rotary table



K pp

x

PC with DSP board

X ref Jerk limited acceleration process

Servo amplifiers

Fig. 1. Experimental apparatus

Y

X

 

TrefX

 

K vi s

Y-axis controller

K pp

y Fluctuation model

K vp

TmX

K ff ( s )

Yref

Cref

C

  

  

TmY

S2

K vp

C-axis controller

     K   pp   Signal delay and backlash model S  1

Fig. 2. Block diagram of proposed controller



K vi s

K ff ( s )

T mC

TrefY



T mC

K vp K vi s

TrefC

 

267

Ryuta Sato and Masaomi Tsutsumi / Procedia CIRP 1 (2012) 265 – 270

does not exist, the inaccuracy due to the servo delay can be avoided.

K ff (s )





K pp   

V ref

Zn

2

Zn  s 2

1 s

R

x

Y-axis

X ref

(X,Y) r

C-axis

(X,Y)

Fig. 3. Block diagram with simplified velocity loop Servo delay

on the partial model matching method [9] is applied. The partial model matching method is a systematic method with which to determine PID control parameters through matching the transfer function of a control system with a reference model. This method was proposed by Kitamori in 1979 [10]. In the method, dynamic response of the system depends on the reference model. The partial model matching method can be applied to decide the velocity loop proportional and integral gains Kvp and Kvi. Figure 3 shows the block diagram of the whole feed drive system with a simplified velocity control loop. Where, R is the ball-screw constant [m/rad], and Zn is the response frequency of velocity loop [rad/s]. The response frequency of the velocity loop can approximately be obtained as velocity loop gain Gv [rad/s] represented as Eq. (1). J is the total inertia of the mechanism [kgm2].

Zn # Gv

K vp J

(1)

The positional response should as fast as possible, and the position overshoot has to be avoided. The position loop proportional gain Kpp which gives critical damping response can be derived as Eq. (2) [9]. K pp

1 4 Zn R 27

(2)

For the system with semi closed loop control, the proportional gain Kpp can also be derived from Eq. (2) by substituting R is equal to 1. 3.2. Feed forward controller Figure 4 explains the influence of servo delay on positional errors. Although the influence of servo delay of translational axes does not depend on the position of the axes in the strokes, the influence of servo delay of rotary axis depends on the distance from the rotational center. This means the servo delay of rotary axis much deteriorate the accuracy of machined shapes. To solve the problem, Lo [11] proposed a control scheme for a five-axis machine tool based on coordinate transformation. However, this study proposes the other approach, feed forward controller. If the servo delay

X-axis (X’,Y’)

Rotational center Servo delay

(X’,Y’)

(a) Translational axes

(b) Rotary axis

Fig. 4. Comparison of influence of servo delay on positional error Rotary encoder of motor

Rotary/Linear encoder of table

Servo amp. Delayed signal Controller

Fig. 5. Encoder signal flow

Transfer function between positional command Xref and position x can be obtained as Eq. (3) from the block diagram shown in Fig. 3.

x X ref

( K ff ( s)  K pp ) RZn2 s(s  Zn ) 2  K pp RZn2

(3)

As the result, the transfer function of feed drive controller Kff (s) which can compensate the servo delay can be derived as follows:

K ff ( s)

1 s( s  Zn ) 2 Zn2 R

(4)

The transfer function shown in Eq. (4) is not proper. A 3rd order Butterworth filter with the cut off frequency of Zn is multiplied to Kff (s). The transfer function Kff (s) of each axis becomes same because the dynamic response of velocity control loops is matched by the partial model matching method. As the result, the dynamic response of the axes can be matched and the servo delay can be avoided by the proposed gain tuning method and feed forward controller. 3.3. Signal delay and backlash compensator Figure 5 describes the encoder signal flow in the experimental system. Although signals from linear and rotary encoders for the positional and angular displacement are directly inputted into the controller, the signal from the rotary encoder of motor is not directly inputted, via servo amplifiers. As the result, the encoder

Ryuta Sato and Masaomi Tsutsumi / Procedia CIRP 1 (2012) 265 – 270

268

Angular displacement rad

signal from the motor is delayed compared with the directly inputted signals from linear and rotary encoders.

S 2 CCW WCCW sin( NC ref )½ S 2 CW WCW sin( NC ref ) ¾¿

CW

CCW

eL B

RCT mC

T tC

Time s

Fig. 6. Influence of encoder signal delay

Figure 6 describes the influence of encoder signal delay on the measured results of the angular displacement, under constant rotational velocity of CCW and CW directions. Where, TtC is angular displacement of table [rad], TmC is angular displacement of motor [rad], and T'mC is angular displacement of motor with the signal delay [rad]. L is the signal delay [s] and RC is the reduction ratio. The amount of backlash between motor and table B [rad] can be defined as Eq. (5), and the motion error due to the signal delay eL [rad] can be represented as Eq. (6). B T tC  RCT mC

eL



(5)

LRCTmC (6)

This means that the appeared backlash B' seems to depend on the rotational velocity, as shown in Eq. (7).

Bc

B  LRCTmC 

(7)

Therefore, the influences of the signal delay and backlash are compensated by the one compensation signal S1 in the proposed method. In this study, the compensation signal S1 is generated by Eq. (8) based on the reference rotational velocity instead of the feedback one. A is the inclination parameter.

S1

B  AC ref

(9)

The compensation signal S2 can be generated based on the reference rotational displacement Cref, because the servo delay is compensated by the feed forward controller mentioned above.

c RCT mC

L

worm wheel. WCCW and WCW are the amplitudes of the fluctuation for CCW and CW directions.

3.5. Jerk limited acceleration process design In the machining of complex shapes, it is required to change velocities rapidly. However, it is difficult to achieve rapid change of velocity especially in the rotary axis driven by a worm gear, because the axis has large reduction ratio. Therefore, the acceleration-deceleration process has to be implemented into the controller, to achieve higher synchronous accuracy. For the translational axes, jerk limited trajectory generation method had been proposed by Erkorkmaz and Altintas [12]. We propose a jerk limited acceleration process design method for synchronous motions of translational and rotary axes. Figure 7 describes the proposed design flow of the jerk limited acceleration process. The acceleration and jerk of angular command for C-axis is limited at the first step. These limitations are achieved by insert the additional time steps into the velocity and acceleration commands. The revised angular command can be obtained by two times integration of the revised acceleration profile. Start Angular command of C-axis Differentiation

Velocity

Maximum acceleration limitation Differentiation

Acceleration

Maximum jerk limitation 2 times integrations



(8)

3.4. Rotational fluctuation compensator It had already investigated that the periodical rotational fluctuation of the rotary axis strongly influences the synchronous accuracy, and the frequency of the fluctuation is same as the rotational frequency of worm shaft [2][8]. Since the amplitude of the fluctuation depends on the rotational direction, the compensation signal S2 is also depended on the rotational direction as shown in Eq. (9). Where, N is the number of teeth of the

Coordinate transformation Position and angular commands with jerk limited acceleration process End

Fig. 7. Design flow of jerk limited acceleration process

The positional commands of the translational axes can be obtained based on the revised angular command using coordinate transformations. As the result, position and angular commands for the translational and rotational axes with jerk limited acceleration process can

269

Ryuta Sato and Masaomi Tsutsumi / Procedia CIRP 1 (2012) 265 – 270

be obtained through the proposed design flow. Tool path is completely not changed through the proposed method. This design flow is not carried out in the real time operation, carried out in the previous step of starting the machine motions. 4. Experimental Results

4.1. Non-uniformed simultaneous 3-axis motion To evaluate the effectiveness of proposed compensators, non-uniformed simultaneous 3-axis motion tests are carried out. This is the synchronous motion of X, Y, and C axes, as shown in Fig. 8 [8]. In the motion, the rotational velocity of C-axis Vc [deg/min] is changed 3 times in 360º as shown in Eq. (10), in order to evaluate the dynamic synchronous accuracy. VC

Vr  AV sin( NC ref )

(10)

The reference velocity Vr is set to 1800 deg/min, and the amplitude of velocity changing AV is set to 720 deg/min, respectively. Number of the velocity changes N is set to 3. Although ball-bar system is originally applied to measure the synchronous accuracy in the motion, the synchronous accuracies are obtained from the measured each axis motions by linear and rotary encoders. Figure 9 shows the measured synchronous accuracies of the non-uniformed 3-axis motion. It can be seen that 3 times fluctuations due to the velocity changing and short To spindle

Z

4.2. Simultaneous 2-axis motion of X and C axes Simultaneous 2-axis motion had been proposed by Tsutsumi et al. to evaluate the dynamic synchronous accuracy of translational and rotary axes [7]. This is a kind of circular test in the polar coordinate system of Caxis, and the velocity of X and C axes are rapidly changed to achieve constant peripheral velocity. Schematic diagram of the 2-axis motion test is shown in Fig. 11. Figure 12 shows the measured results of the simultaneous 2-axis motion. Figure (a) shows the result without compensators, and (b) shows the result with the proposed compensators. The peripheral velocity is set to 1000 mm/min. It can be seen that large tracking error around 180º in figure (a). Cause of this error is the servo delay of X and

XY

Ball bar

XY

cycle fluctuations due to the worm gear are appeared. In addition, the amplitude of the longer cycle fluctuation depends on the rotational direction. This phenomenon is caused from the signal delay of the rotary encoder of Caxis motor. Figure 10 shows the measured synchronous accuracy with the proposed compensators. It is clear from the figure that the long and short fluctuations can effectively be compensated by the proposed compensators, excluding the spike like errors. Cause of these spiky errors is quadrant glitches of translational axes. Hence it is expected that the spiky error can also be compensated by the friction compensator [13].

Rotary table

Ball bar C

50 mm

Ti

C

C

O

X

Y

Pi+1

50 2 mm

Pi OT

Ii

A

r (5

P0

Deviation Pm

0

Fig. 11. Schematic diagram of simultaneous 2-axis motion

-5

CCW 2

4

6 8 Time s

10

12

90º

Fig. 9. Measured synchronous accuracy of non-uniformed simultaneous 3-axis motion without compensators Deviation Pm

Cutter path

XT (55 mm)

X

CW

5

-10 0

Y

m)

10

0m

Fig. 8. Schematic diagram of measurement method for simultaneous 3axis motion

10

1 div.: 5Pm 180º

CW

5

90º

1 div.: 5Pm

0º

0º

180º

0 -5 -10 0

CCW 2

4

6 8 Time s

10

12

Fig. 10. Measured synchronous accuracy of non-uniformed simultaneous 3-axis motion with proposed compensators

270º

(a) Without compensators

270º

(b) With proposed compensators

Fig. 12. Measured results of simultaneous 2-axis motion

Ryuta Sato and Masaomi Tsutsumi / Procedia CIRP 1 (2012) 265 – 270

270

is confirmed that the proposed methods can improve the dynamic synchronous accuracies.

End mill

Feed direction

Workpiece

Acknowledgement Z

r5

B

This work was supported by Japan Society for the Promotion of Science (JSPS), Grant-in-Aid for Young Scientists (B), No.18760164.

100 X

Rotational center

Fig. 13. Schematic diagram of ellipse shape machining motion

References Nominal: 2500Pm/div Deviation: 50Pm/div

(a) Without compensators

Nominal: 2500Pm/div Deviation: 50Pm/div

(b) With proposed compensators

Fig. 14. Measured results of ellipse shape machining motion

C axes because the velocity of the axes is rapidly changed around 180º. Short cycle fluctuation also exists on the trajectory. On the other hand, the synchronous errors cannot be observed on the results with the proposed compensators as shown in figure (b), excluding the spiky errors due to quadrant glitches of X-axis. 4.3. Ellipse shape machining motion Figure 13 describes the ellipse shape machining motion using X, Z, and B axes. Simultaneous 3-axis motions are required around both edges. Y and C axes are controlled instead of Z and B axes in the experiment. Figure 14 shows the experimental results of tool path profile of the motion around left edge. The results are obtained from the measured positional and rotational displacements of each axis. Figure (a) shows the results without compensators, and figure (b) shows the results with the proposed compensators. It can be seen from the results that although large tracking error exists due to the servo delay in the result without compensators (figure (a)), the error cannot be observed in the result with the proposed compensators (figure (b)). 5. Conclusions

In this study, motion control techniques consists of servo gain tuning method, feed forward controller, signal delay and backlash compensator, rotational fluctuation compensator, and jerk limited acceleration process design are proposed. As the results of the experiments, it

[1] Kakino Y, Ihara Y, Sato K, Otsubo H. A study on the motion accuracy of NC machine tools (7th report) -the measurement of motion accuracy of 5-axis machine by DBB test. Journal of JSPE 1994; 60, 5, p. 718-722 (in Japanese) [2] Tsutsumi M, Saito A. Identification and compensation of systematic deviations particular to 5-axis machining centers. International Journal of Machine Tools & Manufacture 2003; 43, p. 771-780 [3] Tsutsumi M, Saito A. Identification of angular and positional deviations inherent to 5-axis machining centers with a tiltingrotary table by simultaneous four-axis control movements. International Journal of Machine Tools & Manufacture 2004; 44, p. 1333-1342 [4] Weikert S. When five axes have to be synchronized. Proceedings of the 7th LAMDAMAP conference 2005; p. 87-96 [5] Weikert S. R-test, a new device for accuracy measurement on five axis machine tools. Annals of the CIRP 2004; 53, 1, p. 429-432 [6] Bringmann B, Knapp W. Model-based 'chase-the-ball' calibration of a 5-axes machining center, Annals of the CIRP 2006; 55, 1, p. 531-534. [7] Tsutsumi M, Yumiza D, Utsumi K. Sato R. Evaluation of synchronous motion in five-axis machining centers with a tilting rotary table. Journal of advanced mechanical design, systems, and manufacturing 2007; 1, 1, p. 24-35 [8] Sato R, Tsutsumi M. Dynamic synchronous accuracy of translational and rotary axes. International journal of mechatronics and manufacturing systems 2011; 4, Nos. 3/4, p. 201-219 [9] Sato R, Tsutsumi M. Modelling, and controller tuning techniques for feed drive systems. Proceedings of the ASME, dynamic systems and control division, Part A 2005; DSC-74-1, p. 669-679 [10] Kitamori T. A design method for control system based upon partial knowledge about controlled processes. Transactions of the Society of Instrument and Control Engineers 1979; 15, p. 549-555 (in Japanese) [11] Lo C.C. Control scheme for a five-axis machine tool based on coordinate transformation. JSME International Journal, Series C 1997; 40, 3, p. 477-483 [12] Erkorkmaz K, Altintas Y. High speed CNC system design. Part I: Jerk limited trajectory generation and quintic spline interpolation. International Journal of Machine Tools & Manufacture 2001; 41, p. 1323-1345 [13] Sato R, Tsutsumi M. Generation mechanism of quadrant glitches and compensation for it in feed drive systems of NC machine tools. International Journal of Automation Technology 2012; 6, 2, p. 154-162